Stable limit theorem for U-statistic processes indexed by a random walk
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Date
2012-12-17
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Abstract
Let (Sn)n2N be a random walk in the domain of attraction of
an a -stable Lévy process and ( (n))n2N a sequence of iid random variables
(called scenery). We want to investigate U-statistics indexed by the random
walk Sn, that is Un :=
P
1 i<j n h( (Si); (Sj )) for some symmetric bivariate
function h. We will prove the weak convergence without the assumption of
finite variance. Additionally, under the assumption of finite moments of order
greater than two, we will establish a law of the iterated logarithm for the
U-statistic Un.
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Keywords
law of the iterated logarithm, random scenery, random walk, stable limits, U-statistics